Convergence of general inverse \(\sigma_{k}\)-flow on Kähler manifolds with Calabi ansatz (Q2855940)

In this paper, the authors study the convergence behaviour of the \(\sigma_k\) flow which is a natural generalization of the \(J\)-flow. It is known that the \(\sigma_k\) flow exists for all time and that it converges to a critical metric under suitable assumptions. Thus, it will be interesting to know the limiting behaviour of the \(\sigma_k\) flow in the general case. Using the Calabi ansatz, the authors study the limiting behaviour of the \(\sigma_k\) flow on \(\mathbb{P}^n\#\overline{\mathbb{P}^n}\) and \(X_{m,n}\) which is a projective bundle over \(\mathbb{P}^n\) of total dimension \(m+n+1\). An application of their work, as pointed out later by Lejmi and Székelyhidi, is to give a counter example of a conjecture due to Donaldson.